Average Error: 14.8 → 0.1
Time: 2.2s
Precision: 64
\[\frac{1}{x + 1} - \frac{1}{x - 1}\]
↓
\[\frac{1}{x + 1} \cdot \frac{-2}{x - 1}\]

Error

Bits error versus x

Derivation

  1. Initial program 14.8

    \[\frac{1}{x + 1} - \frac{1}{x - 1}\]
  2. Using strategy rm

  3. Applied frac-sub14.2

    \[\leadsto \color{blue}{\frac{1 \cdot \left(x - 1\right) - \left(x + 1\right) \cdot 1}{\left(x + 1\right) \cdot \left(x - 1\right)}}\]

  4. Simplified14.2

    \[\leadsto \frac{\color{blue}{1 \cdot \left(\left(x - 1\right) - \left(x + 1\right)\right)}}{\left(x + 1\right) \cdot \left(x - 1\right)}\]

  5. Simplified14.2

    \[\leadsto \frac{1 \cdot \left(\left(x - 1\right) - \left(x + 1\right)\right)}{\color{blue}{x \cdot x - 1 \cdot 1}}\]

  6. Taylor expanded around 0 0.4

    \[\leadsto \frac{1 \cdot \color{blue}{\left(-2\right)}}{x \cdot x - 1 \cdot 1}\]
  7. Using strategy rm

  8. Applied difference-of-squares0.4

    \[\leadsto \frac{1 \cdot \left(-2\right)}{\color{blue}{\left(x + 1\right) \cdot \left(x - 1\right)}}\]

  9. Applied times-frac0.1

    \[\leadsto \color{blue}{\frac{1}{x + 1} \cdot \frac{-2}{x - 1}}\]

  10. Final simplification0.1

    \[\leadsto \frac{1}{x + 1} \cdot \frac{-2}{x - 1}\]

Reproduce

herbie shell --seed 2020130 
(FPCore (x)
  :name "Asymptote A"
  :precision binary64
  (- (/ 1.0 (+ x 1.0)) (/ 1.0 (- x 1.0))))